I've seen some questions discussing the capitalization of "queen" in general, but I'm wondering if you would ever capitalize the "my" in "my queen," particularly in a line of dialogue such as "Very well, my queen" or "It will be done, my queen." The queen being referenced is a fictional one of a fictional country.
Given that the speaker could use their intonation/emphasis to differentiate their own use of "My Queen," as a title and "my queen," as a common noun, differing capitalization could lend the same nuance to the written dialogue.
When writing fiction, it could also be useful to imply a character's loyalty, respect, or reverence to this queen, by capitalizing the title in their reported speech. This obviously comes much closer to a stylistic decision than a strictly functional grammatical decision. For your most careful readers, though, this detail could add another layer to characters that would otherwise need gratuitously overt declarations of their loyalties — which could disrupt the flow of the passage or just seem ham-fisted.
You may or may not, depending on the context. My Queen is a definite title or a mark of respect, and is usually used by utterly loyal servants to address their female master. As for my Queen, it is usually used whenever it is usually used by commoner to address their master or for higher ranking servants of the queen to address her. But, take note that often, Middle English and plays often use "My Queen" and "my Queen" is used in more modern productions.
P.S. This is my first answer. Comment if it's wrong
If the question relates specifically to dialogue (and it does), then the choice to capitalize or not resides completely with the author. It's a stylistic choice, and an editor would only insist on consistency. I.e., "...my queen" in all instances, or "...My Queen" in all instances.
See: this page, under "Language" C11.
I would capitalize "My Queen" because it is a title. You would capitalize "Mom" or "Dad" for the same reason. You however would not capitalize "my mom" or "my dad" because that is who they are. "My Queen" is tricky because it's both a title and describing who the person but you would capitalize "My Majesty" therefore you would capitalize "My Queen".